Sliding the printer to the side, or lifting and GENTLY placing it elsewhere should not change anything.
Putting it back in the original spot on of the bench for printing should be fine.
Printing from a different spot might affect the resonance frequencies of vibrations and input shaping. Repeat the printer calibration if you are not happy with the results in a new spot.
Auto-levelling is a misnomer. The bed height is adjusting to maintain a constant distance between the bed and the nozzle at all points. Again, it should not not be affected by moving the printer unless you are handling the printer roughly enough to twist the frame.